\begin{section}{Constructing an Analogue to the Craighero-Gattazzo Surface}

In this section we experiment with the Craighero-Gattazzo quintic by placing different restrictions on the coefficients.

Let $\mathcal{F}$ be the equation for the quintic. After dehomogenizing with $t = 1$ and making the quadratic part into the square of a linear term, we get

\begin{equation}
\mathcal{F}_{3} = \alpha y^3 + \beta y^{2}z + \delta yz^{2} + \gamma z^3.
\end{equation}

We now set $\alpha = \beta = \delta = 0.$

\begin{equation}
\alpha = -a^2m^3 + bm^2 - cm + m^2 = 0.
\end{equation}

\begin{equation}
\beta = -3a^3m^2 + 2abm + cm^2 - ac - fm + c = 0.
\end{equation}

\begin{equation}
\delta = -3a^4m + a^2b + 2aem - af - bm + c = 0.
\end{equation}

These restrictions on the coefficients of $\mathcal{F}$ give

\begin{equation}
\mathcal{F}_{3} = \gamma z^3  = (-a^5 + a^2e - ab + 1)z^3.
\end{equation}

We then blowup the surface in the direction of $y$ using the linear change of coordinates $$(x,y,z) \to (xy, y, zy)/y^2.$$

After the blowup, we examine $\mathcal{F}_{2}$ and notice that the coefficient of the $yz$ term is exactly $\beta$, which vanishes and allows us to write

\begin{equation}
\mathcal{F}_{2} = (x + \mu y)^2 = x^2 + 2\mu xy + \mu^2 y^2.
\end{equation}

A quick calculation shows that $$\mu = \frac{3a^{2}m^{2} - 2bm + c}{2}$$.

Then we make the substitution $$(x,y) \to (-y\mu, y) \implies \mathcal{F}_{2} = 0,$$ which gives a fourth restriction

\begin{equation}
-9a^4m^4 + 12a^2bm^3 - 6acm^2 - 4b^2m^2 - 8am^3 + 4bcm - 8am^2 + 4em^2 - c^2 = 0.
\end{equation}

Note that this is equivalently a condition that the ``discriminant'' vanishes,

\begin{eqnarray}
\nabla & = & (D[xy, \mathcal{F}_2])^2 -( D[x^2, \mathcal{F}_2])(D[y^2, \mathcal{F}_2]) = 0 \\
& = & \nabla y^2 + 4\beta yz = 0.
\end{eqnarray}

Now we have $\mathcal{F}_{2} = 0$ and $$\mathcal{F}_{3} = \tilde{\alpha}z^3 + \tilde{\beta}yz^2 + \tilde{\delta}y^2z.$$

Where,

\begin{multline}
\tilde{\alpha} = -27a^{11} + 45a^8e - 18a^7b - 36a^7m - 24a^5e^2 + 12a^5c + 18a^4be + 24a^4em - 3a^3b^2 + 4a^2e^3 - 12a^4m \\ 
- 12a^3bm - 4a^3m^2 - 8a^2ce - 4abe^2 + 4a^2m^2 + 4abc + b^2e + 8aem - 4bm. 
\end{multline}

\begin{multline}
\tilde{\beta} = -36a^7m + 12a^5b + 36a^4em - 6a^4f - 12a^3bm - 24a^3m^2 - 8a^4 - 8a^2be - 8ae^2m + 4ab^2 + 4aef + 8acm \\
 + 4bem - 4af - 2bf - 8m^2 + 8a.
\end{multline}

\begin{equation}
\tilde{\delta} = 4\delta =  (4)(-3a^3m^2 + 2abm + em^2 - ac - fm + c) = 0.
\end{equation}

Hence,

\begin{equation}
\mathcal{F}_{3} = \tilde{\alpha}z^3 + \tilde{\beta}yz^2.
\end{equation}

\end{section}


\begin{section}{Restrictions on the Coefficients for Analogue to Craighero-Gattazzo surface}

These equations are restrictions on the coefficients for the an analogue to the Craighero-Gattazzo surface where $\mathcal{F}_{3} = \gamma z^3$ and the blowup is given by the analytic change of coordinates $$(x,y,z) \to \frac{(x*y,y,z*y)}{y^2}$$ working in local coordinate system where $t = 1$.

\begin{equation}
\boxed{-a^2m^3 + bm^2 - cm + m^2 = 0.}
\end{equation}

\begin{equation}
\boxed{-3a^3m^2 + 2abm + cm^2 - ac - fm + c = 0.}
\end{equation}

\begin{equation}
\boxed{-3a^4m + a^2b + 2aem - af - bm + c = 0.}
\end{equation}

\begin{equation}
\boxed{-9a^4m^4 + 12a^2bm^3 - 6acm^2 - 4b^2m^2 - 8am^3 + 4bcm - 8am^2 + 4em^2 - c^2 = 0.}
\end{equation}

\begin{equation}
\boxed{3a^3m^4 - 3a^3m^3 - 4abm - a^2m^2 + 3acm^2 + bem^2 + acm - cem + m^3 = 0.}
\end{equation}

\begin{equation}
\boxed{\begin{split}
-18a^4m^3 + & 18a^3bm^2 + 6a^2em^3 - 6a^3cm - 3a^2fm^2 - 4am^4 - 4ab^2m - 12a^2m^2 - 4bem^2 + 2abc \\
& - 4a^2m + 4aem + 2cem + 2bfm + 2fm^2 - cf - 2fm + 4m = 0.
\end{split}}
\end{equation}

\end{section}
